In quadrilateral \(ABCD,\) \(AB = AD = 1,\) \(BD = \sqrt{2},\) \(BC = \dfrac{ \sqrt{2} }{ 2 },\) and \(CD = \dfrac{ \sqrt{6} }{ 2}.\) The area of triangle \(ABC\) can be expressed in the form \( \dfrac{\sqrt{a} + b}{c} \), where \(a, \) \(b, \) and \(c\) are positive integers and \(a\) is square-free. What is the value of \(a+b+c\)?

This problem is posed by Michael T.

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